Higher-Dimensional Spaces

The intuitions that students accumulate in dealing with coordinate pairs in
the plane and coordinate triples in three-dimensional space lead naturally
to coordinate geometry in higher dimensions. A thorough understanding of
two and three dimensions provides an important foundation for the powerful
generalizations of vector and matrix algebra in science and engineering, in
economics and social science, and especially computer science and graphics.
We illustrate this progression with two examples.

The vertices of a square can be given by four points (0,0), (1,0), (1, 1),
and (0, 1). To obtain the vertices of a cube, we can take the points of a
zero in the third coordinate and then move the square one unit in the third
direction to obtain four more vertices, with a 1 in the last coordinate:

(0, 0, 0),

(1, 0, 0),

(1, 1, 0),

(0, 1, 0),

(0, 0, 1),

(1, 0, 1),

(1, 1, 1),

(0, 1, 1).

Thus we can describe either the square or the cube as having vertices that
are either 0 or 1 in each coordinate.

Figure 32. Generalizing the Pythagorean theorem to three
dimensions by applying it to two different triangles found in a rectangular
box.

The procedure generalizes automatically: to obtain the vertices of a
hypercube, we start with the eight vertices of a cube and put 0 in the
final coordinate and then "move the cube in a fourth direction" to obtain
eight more points with 1 the last coordinate:

(0,0,0,0),

(1,0,0,0),

(1,1,0,0),

(0,1,0,0),

(0,0,1,0),

(1,0,1,0),

(1,1,1,0),

(0,1,1,0),

(0,0,0,1),

(1,0,0,1),

(1,1,0,1),

(0,1,0,1),

(0,0,1,1),

(1,0,1,1),

(1,1,1,1),

(0,1,1,1).

We thus obtain the sixteen vertices of a hypercube, with 0 or 1 in each of
four coordinates. It is this sort of representation that is ideal for
communicating with a computer.

A second topic that generalizes in a very nice way is the Pythagorean
theorem. If we think of this theorem as a way of calculating the length of
the diagonal of a rectangle with given sides, then the extension to three
dimensions is immediate: given a solid bounded by rectangular sides, we
first apply the theorem to one side and then apply it to a rectangle built
over the first diagonal (Figure 32). We easily get e2 =
c2+d2 =
c2+ (a2+b2), so the length of
the diagonal of a rectangular prism with sides a, b, and
c is (a2 + b2 +
c2)1/2. The pattern is established, and
the distance formula in four-dimensional space follows almost immediately.
Students can then calculate the lengths of diagonals of the hypercube with
the 0-1 coordinates. It turns out that the length of the major diagonal of
a four-dimensional cube — say from (0,0,0,0) to (1,1,1,1) — is
41/2 = 2which is twice the length of a side.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.